Physical and Geometrical Interpretation of Fractional Derivatives in Viscoelasticity and Transport Phenomena

Abstract

Few could have imagined the vast developments made in the field of fractional calculus which was first merely mentioned in the year 1695 in a correspondence between the pioneers of calculus, Leibniz and L’Hôpital. However, since fractional order derivatives can easily be seen as a “natural” generalization of the integer order derivatives, their studies have mostly been limited to the mathematics community. This is evident from the fact that despite having more than three hundred years of history, the order of the fractional derivative is mostly obtained by curve-fitting the experimental data from the theoretically predicted curves. Besides, very few works have demonstrated a deductive approach to fractional derivatives from the real physical processes. The physical interpretation of the fractional order has remained an open question for the fractional community as well as for those who use them to describe anomalous, complex, and memory-driven physical phenomena. Consequently, despite its widespread applications in the field of acoustics, rheology, seismology, and medical physics, fractional calculus has been plagued as an “empirical only” mathematical methodology. This thesis mainly aims at identifying the mechanisms which give rise to power law behavior, and hence a deductive approach to the fractional derivatives. This is first achieved by mapping the grain-shearing model of wave propagation in marine sediments into the framework of fractional calculus. The time-dependent viscosity of pore-fluid in the model is identified as the non-Newtonian property of rheopecty. The wave equations from the grain-shearing model turn out to be of fractional order. In the pursuit to analyze the grain-shearing model, it is found that a linearly time-varying Maxwell model yields a similar relaxation modulus as that of a fractional dashpot, and Lomnitz’s law as its creep compliance. Further, a linearly time-varying viscosity whose varying part dominates over the constant part is established as the common physical mechanism underlying both Nutting’s law and Lomnitz’s law. The fractional order in the Nutting law, and also the terms in the creep law gain physical interpretation. This physical justification has been lacking in both Nutting’s law and Lomnitz’s law since their inception in 1921 and 1956 respectively. We have also investigated fluid dynamics in fractal media. It is shown that the fractional Navier-Stokes equation and the fractional momentum diffusion equation naturally arise in such a medium. These findings infer that fluid flow in fractal media lead to fractional derivatives in time, the order of which is related to the fractal dimension of the medium. Thus, fractional derivatives also gain a form of geometrical interpretation. In addition, spatial dispersion of elastic waves in a nonlocal elastic bar is studied using space-fractional derivatives. The nonlocal attenuation kernel is tempered in order to circumvent the strong singularity encountered at the local point of stress application. Though the dispersion behavior obtained numerically is unusual, it turns out to be physically reasonable. The overall goal of this thesis is to show that fractional calculus is not just a mathematical framework which can only be empirically introduced to curve-fit the experimental observations. Rather, it has an inherent connection to real physical processes which needs to be explored more. We hope that the results obtained here may benefit the scientific communities of fractional calculus, seismology, non-Newtonian rheology, sediment acoustics, and fluid dynamics.